7/28/2019 Tutorial 06 Axisymmetric
http://slidepdf.com/reader/full/tutorial-06-axisymmetric 1/24
Axisymmetry Tutorial 6-1
Phase2 v.8.0 Tutorial Manual
Axisymmetry Tutorial
This tutorial will illustrate the axisymmetric modeling option of Phase2 .
Axisymmetric modeling allows you to analyze a 3-D excavation which is
rotationally symmetric about an axis. The input is 2-dimensional, but the
analysis results apply to the 3-dimensional problem.
An Axisymmetric model in Phase2 is typically used to analyze the end of
a circular (or nearly circular) tunnel. The model we will be analyzing,
shown above, represents the end of a cylindrical tunnel of 4 meter radius.
The finished product of this tutorial can be found in the Tutorial 06
Axisymmetric.fez file. All tutorial files installed with Phase2 8.0 can be
accessed by selecting File > Recent Folders > Tutorials Folder from the
Phase2 main menu.
A few representations of simple axisymmetric models are shown below.
28 , 18 0 , 18
0 , 12
0 , 6
0 , 0 4 , 0
4 , –24 12 , –24 20 , –24 28 , –24
user-defined external boundary
10 MPa
10 MPa
x =0(axis of
symmetry)
7/28/2019 Tutorial 06 Axisymmetric
http://slidepdf.com/reader/full/tutorial-06-axisymmetric 2/24
Axisymmetry Tutorial 6-2
Phase2 v.8.0 Tutorial Manual
a) sphere b) cylinder
c) ‘open’ cylinder d) infinite cylinder
Figure 6-1: Simple axisymmetric models. For a), b) and c), the left edge of each
boundary is coincident with the X = 0 (vertical) axis. For d), the boundary is displacedfrom the X = 0 axis, therefore modeling an infinite circular tunnel.
NOTE:
• Only an external boundary is necessary to define an axisymmetric
model – the excavation is implicitly defined by the shape and
location (relative to the x=0 axis) of the external boundary.
Appropriate boundary conditions must also be applied to complete
the modeling.
• The axis of rotation is always the X = 0 (vertical) axis. Your model
must always be mapped to fit this convention, regardless of the
actual orientation of the excavation. Because of the symmetry,
only “half” of the problem needs to be defined.
x =0 x =0
x =0
x =0
7/28/2019 Tutorial 06 Axisymmetric
http://slidepdf.com/reader/full/tutorial-06-axisymmetric 3/24
Axisymmetry Tutorial 6-3
Phase2 v.8.0 Tutorial Manual
So, to visualize an axisymmetric excavation, just imagine the shape
formed by rotating the external boundary about the x=0 axis. Note that
Figure 6-1 is for illustration, and that actually the boundaries should be
extended relative to the excavations (in Figures 6-1a and 6-1b) to ensure
that the fixed boundary conditions do not affect the results around the
excavation. Figure 6-1d can actually be defined by a narrow horizontal
strip, since the problem is effectively one-dimensional (i.e. results willonly vary along a line perpendicular to the tunnel), and is in fact
equivalent to a circular excavation in a plane strain analysis.
There are various restrictions on the use of axisymmetric modeling in
Phase2 , for example:
• the field stress must be axisymmetric i.e., aligned in the axial and
radial directions.
• cannot be used with BOLTS (however LINERS are permitted)
• cannot be used with JOINTS
• all materials must have ISOTROPIC elastic properties
In this tutorial, we will look at results not only around the end of the
tunnel, but also along its length, where the conditions are effectively
plane strain. We will later verify these results by comparing with a plane
strain analysis.
7/28/2019 Tutorial 06 Axisymmetric
http://slidepdf.com/reader/full/tutorial-06-axisymmetric 4/24
Axisymmetry Tutorial 6-4
Phase2 v.8.0 Tutorial Manual
Model
Start the Phase2 Model program.
Project Settings
When you are creating an axisymmetric model, the first thing you should
always do is set the Analysis Type to Axisymmetric in the Project
Settings dialog.
Select: Analysis→ Project Settings
In the Project Settings dialog, toggle the Analysis Type to Axisymmetric,and select OK.
Entering Boundaries
Since only an external boundary is required to define an axisymmetric
problem in Phase2 , proceed directly to the Add External option (rather
than the usual procedure of first adding excavations).
Select: Boundaries→ Add External
Enter the following coordinates at the prompts:
Ent er ver t ex [ t =t abl e, i =ci r cl e, esc=cancel ] : 28 18 Ent er vert ex [ . . . ] : 0 18 Ent er vert ex [ . . . ] : 0 12 Ent er vert ex [ . . . ] : 0 6
Ent er vert ex [ . . . ] : 0 0 Ent er vert ex [ . . . ] : 4 0
Ent er vert ex [ . . . ] : 4 -24
Ent er vert ex [ . . . ] : 12 -24
Enter:
Project Name =(optional)
Number of Stages =1 Analysis =Axisymmetric
Max. #of iterations =500
Tolerance =0.001
#Load Steps =Auto
Solver Type =Gauss. Elim.
Units =Metric (MPa)
7/28/2019 Tutorial 06 Axisymmetric
http://slidepdf.com/reader/full/tutorial-06-axisymmetric 5/24
Axisymmetry Tutorial 6-5
Phase2 v.8.0 Tutorial Manual
Ent er vert ex [ . . . ] : 20 -24 Ent er vert ex [ . . . ] : 28 -24 Ent er ver t ex [ . . . , c=cl ose, esc=cancel ] : c
Select Zoom All (or press F2) to zoom the model to the center of the view.
This is the only boundary required for the problem, so we can proceed tothe meshing.
Meshing
As usual, we will discretize and mesh the model. However, let’s first take
a look at the Mesh Setup option.
Select: Mesh→ Setup
Notice that the Mesh Setup dialog normally asks you for the # Excavation
Nodes. However, for models which have no explicitly defined Excavation
boundaries (such as this one), the # External Boundary Nodes is enteredinstead. Also, the Gradation Factor is not applicable when there are no
Excavation boundaries defined. Select OK or Cancel, since we are using
the default parameters.
Now let’s discretize the external boundary.
Select: Mesh→ Discretize
The status bar will indicate the actual number of discretizations created
on the external boundary.
Di scret i zat i ons: External =60
Note that this is a fairly coarse discretization. The boundary segments
which are part of, or adjacent to, the excavation, will require a finer
discretization. We can do this with the Custom Discretize option.
Enter:
Mesh Type =Graded
Elem. Type =3 Noded Tri.
#External Nodes =60
7/28/2019 Tutorial 06 Axisymmetric
http://slidepdf.com/reader/full/tutorial-06-axisymmetric 6/24
Axisymmetry Tutorial 6-6
Phase2 v.8.0 Tutorial Manual
Custom Discretization
Select: Mesh→ Custom Discretize
Sel ect Segment s t o Di scr et i ze [ ent er=done, esc=cancel ] : use the
mouse to select the long edge of the tunnel ie. the long
vertical segment at the lower left of the external boundary.
Right-click and select Done Selection, or just press Enter.
In the Custom Discretize dialog, enter 60 as the number of discretizations,
and select OK.
The length of the tunnel is now discretized into 60 elements. Now follow
this same procedure to apply custom discretizations as indicated in the
margin figure.
Note:
• you can select more than one line segment at a time, if they
require the same number of discretizations (for example, thesegments with 6 and 12 discretizations, in this case).
• the segments which are not marked in the margin figure, are to
be left at their original discretizations.
Mesh
Now select the Mesh option from the toolbar or the Mesh menu, and the
mesh will be generated, based on the discretization you just created.
Select: Mesh→Mesh
The status bar will indicate the total number of elements and nodes in
the mesh.
NODES = 397 ELEMENTS = 648
At this point, we will make the following observation – you may have
wondered, when we created the external boundary, why we added the
extra vertices on the upper left vertical segment and lower right
horizontal segment of the boundary, since these boundaries could have
been defined by single segments. As you can now see, the extra segments
allowed us to custom discretize the boundaries, in order to get a smooth
transition between the fine mesh around the tunnel, and the coarser
mesh of the rest of the boundary. (If we did not do this, a poor meshwould be generated where the fine to coarse transition is too abrupt.)
CUSTOM DISCRETIZATION
7/28/2019 Tutorial 06 Axisymmetric
http://slidepdf.com/reader/full/tutorial-06-axisymmetric 7/24
Axisymmetry Tutorial 6-7
Phase2 v.8.0 Tutorial Manual
Boundary Conditions
In most of the tutorials so far, we have not been specifying any boundary
conditions. We were using the default boundary condition, which is a
fixed (zero displacement) external boundary.
For an axisymmetric model, the external boundary conditions are veryimportant, and must be user specified. We cannot simply leave the
boundary fixed, or else nothing would happen (i.e., no displacements
could take place).
Figure 6-2: Displacement boundary conditions for axisymmetric model.
First, let’s ‘free’ the tunnel boundaries.
Select: Displacements→ Free
Sel ect boundar y segment s t o f r ee [ ent er=done, esc=cancel ] : Use
the mouse to select the 2 segments marked FREE in Figure 6-2. When finished, right-click and select Done Selection, or press
Enter.
The triangular pin symbols are now gone from the two boundary
segments (representing the end of the tunnel and the length of the
tunnel), indicating that they are free to move with no restriction in anydirection.
Now let’s specify the boundary segments at the upper left edge as
restrained in the X direction, but free to move in the Y direction. (These
segments are located on the axis of symmetry, and therefore must have
zero X displacement).
7/28/2019 Tutorial 06 Axisymmetric
http://slidepdf.com/reader/full/tutorial-06-axisymmetric 8/24
Axisymmetry Tutorial 6-8
Phase2 v.8.0 Tutorial Manual
Select: Displacements→ Restrain X
Sel ect boundary segment s t o rest r ai n i n X di r ect i on[ ent er=done, esc=cancel ] : Use the mouse to select the three
segments marked FIXED X in Figure 6-2. Right-click and select
Done Selection, or press Enter.
Observe that the triangular pins on these segments have been replaced
by vertical rollers.
Now let’s specify the boundary segments along the bottom as restrained
in the Y direction, but free to move in the X direction.
Select: Displacements→ Restrain Y
Sel ect boundary segment s t o rest r ai n i n Y di r ect i on[ ent er=done, esc=cancel ] : Use the mouse to select the three
segments marked FIXED Y in Figure 6-2. Right-click and select
Done Selection, or press Enter.
Observe that the triangular pins on the bottom segments have been
replaced by horizontal rollers.
Now we have a bit of tidying up to do.
Select: Displacements→ Restrain X,Y
1. Right-click the mouse and select Pick by Boundary Nodes from
the popup menu. This will change the mode of restraint
application from boundary segments to boundary nodes. (The
mode can also be changed in the Selection Mode sub-menu of the
Displacements menu).
2. Select the upper left corner of the model, i.e. the vertex at (0 , 18).
3. Select the lower right corner of the model, i.e. the vertex at (28 , -
24).
4. Right-click and select Done Selection.
5. A triangular pin symbol should now replace the roller symbol at
these two vertices.
The above steps were necessary, since the upper left and lower right
vertices required a Restrained XY condition. This leads us to an
important point – after applying restraints to boundary segments, youshould always check that nodes at the ends of segments have the correct
conditions applied.
TIP: restraints can also be applied directly with a right-click shortcut, by
right-clicking on segments or vertices and selecting a restraint option
from the popup menu.
7/28/2019 Tutorial 06 Axisymmetric
http://slidepdf.com/reader/full/tutorial-06-axisymmetric 9/24
Axisymmetry Tutorial 6-9
Phase2 v.8.0 Tutorial Manual
Field Stress
For this example, we will be using the default hydrostatic stress field of
10 MPa. However, let’s look at the field stress option, to see how an
axisymmetric field stress is specified.
Select: Loading→ Field Stress
Only two independent principal stresses (Horizontal and Vertical) are
specified for an axisymmetric problem, and no angle is allowed. Select OK
or Cancel.
Note the following correspondences between Plane Strain and
Axisymmetric field stress, as defined in Phase2 :
PLANE STRAIN AXISYMMETRIC
Sigma 1 (in-plane) ‘Horizontal’ stress
Sigma 3 (in-plane) ‘Horizontal’ stress
Sigma Z (out-of-plane) ‘Vertical’ stress
Angle not applicable
Table 6-1: Equivalent plane strain and axisymmetric field stress components.
NOTE:
• The Horizontal (axisymmetric) field stress can also be thought of
as a uniform radial stress around the excavation.
• An angle cannot be specified for the axisymmetric field stress,because this would require a true 3-dimensional analysis, which
is beyond the scope of the Phase2 axisymmetric analysis.
7/28/2019 Tutorial 06 Axisymmetric
http://slidepdf.com/reader/full/tutorial-06-axisymmetric 10/24
Axisymmetry Tutorial 6-10
Phase2 v.8.0 Tutorial Manual
• It should be emphasized that the terms Horizontal and Vertical
refer strictly to the setup of your model in Phase2 , and not
necessarily the true orientation of your excavation. The Vertical
stress is the stress in the axial direction (i.e. the axis of rotational
symmetry), and the Horizontal stress is the field stress
perpendicular to this axis.
Properties
In this tutorial, we will not be defining or assigning any properties,
therefore all default properties will be in effect. We have dealt with
defining and assigning properties in previous tutorials.
For reference purposes, the default rock properties which will be in effect
are shown in the margin. (If you want, you can verify this by selecting
Properties→ Define Materials).
Note that our analysis will therefore be elastic. Also note the values of
Young’s Modulus and Poisson’s ratio.
We have now completed the modeling. Your finished model should appear
as shown below.
Figure 6-3: Finished model –Phase2 Axisymmetric Tutorial
Enter:
Name =Material 1
Init.El.Ld.=Fld Stress Only
Material Type =Isotropic
Young’s Modulus =20000
Poisson’s Ratio =0.3
Failure Crit. =Mohr Coul.
Material Type =Elastic
Tens. Strength =0Fric. Angle (peak) =35
Cohesion (peak) =10.5
7/28/2019 Tutorial 06 Axisymmetric
http://slidepdf.com/reader/full/tutorial-06-axisymmetric 11/24
Axisymmetry Tutorial 6-11
Phase2 v.8.0 Tutorial Manual
Compute
Before you analyze your model, save it as a file called axi1.fez.
Select: File→ Save
Use the Save As dialog to save the file. You are now ready to run the
analysis.
Select: Analysis→ Compute
The Phase2 Compute engine will proceed in running the analysis. When
completed, you will be ready to view the results in Interpret.
Interpret
To view the results of the analysis:
Select: Analysis→ Interpret
This will start the Phase2 Interpret program.
Sigma 1
On the Sigma 1 contours, notice the stress concentration at the ‘corner’ of
the tunnel (remember the tunnel is circular).
Toggle the stress trajectories on by selecting the Stress Trajectories
button in the toolbar. The principal stress trajectories illustrate the
“stress flow” around the end of the tunnel.
Figure 6-4: Principal stress trajectories around axisymmetric excavation.
7/28/2019 Tutorial 06 Axisymmetric
http://slidepdf.com/reader/full/tutorial-06-axisymmetric 12/24
Axisymmetry Tutorial 6-12
Phase2 v.8.0 Tutorial Manual
The square dot markers in the upper right corner of the model indicate
nodes where the difference between Sigma 1 and Sigma 3 is less than a
certain tolerance, so that the conditions are effectively hydrostatic, and a
distinction between ‘major’ and ‘minor’ principal stress is not warranted.
Toggle the stress trajectories off by re-selecting the Stress Trajectories
toolbar button.
As an optional step, look at Sigma 3 and Sigma Z, and consider the
significance of the principal stress results from an axisymmetric analysis.
As with plane strain, Sigma 1 and Sigma 3 are the major and minor ‘in-
plane’ principal stresses. Sigma Z is therefore the ‘out-of-plane’ stress,
however, since the problem is axisymmetric, Sigma Z is really the
induced circumferential or hoop stress around the excavation.
Displacement
Now let’s view the displacements.
Select:
Note the maximum total displacement displayed in the status bar is
0.00246 m, or just over 2 mm. Although this is quite small, remember
that our analysis was elastic and we used a relatively high Young’s
modulus.
Now let’s view the deformation vectors. Right-click the mouse and select
Display Options.
In the Display Options dialog, toggle Deformation Vectors on, enter a
Scale Factor of 600, and select Done.
The deformation vectors show the inward displacement along the length
and face of the tunnel. Notice how the “corner” of the tunnel effectively
restrains the displacements in both x and y directions.
7/28/2019 Tutorial 06 Axisymmetric
http://slidepdf.com/reader/full/tutorial-06-axisymmetric 13/24
Axisymmetry Tutorial 6-13
Phase2 v.8.0 Tutorial Manual
Figure 6-5: Displacement contours and vectors around axisymmetric excavation.
Toggle the deformation vectors off by selecting the Deformation Vectors
button in the toolbar.
Query Data
Phase2 allows the user to query data anywhere in the rock mass, to
obtain values interpolated from the contour plots. A query can be a single
point, a line segment, or an arbitrary polyline.
Let’s first create a query along the length of the tunnel.
Select: Query→ Add Material Query
1. It will be handy to use the Snap option, so right-click the mouse
and make sure the Snap option is selected. While in Snap mode, if
you place the cursor near a vertex, you can snap exactly to the
location of a vertex.
2. Use the mouse to select the vertex at (4 , 0).
3. Use the mouse to select the vertex at (4 , -24).
4. Right-click the mouse and select Done. You will see the followingdialog:
7/28/2019 Tutorial 06 Axisymmetric
http://slidepdf.com/reader/full/tutorial-06-axisymmetric 14/24
Axisymmetry Tutorial 6-14
Phase2 v.8.0 Tutorial Manual
Toggle off Display Queried Values and Show ID Number and select OK.
A query has now been created along the length of the tunnel. The 50
locations at which data will be generated are indicated by black cross
markers.
Graphing a Query
Graphs are created from queries with the Graph Material Queries option.
However, a convenient shortcut to graph data for a single query, is tosimply right-click on a query and select Graph Data.
1. Right-click on the query you just created (i.e. anywhere along the
length of the tunnel), and select Graph Data from the popup
menu.
2. You will see the Graph Query Data dialog. Select the Plot button
in this dialog.
3. You should see the graph in Figure 6-6. Since we were viewing
the Total Displacement contours, we obtained a graph of total
displacement vs. distance along the query. The data graphed
always corresponds to the contoured data being viewed.
Figure 6-6: Total displacement along length of tunnel.
7/28/2019 Tutorial 06 Axisymmetric
http://slidepdf.com/reader/full/tutorial-06-axisymmetric 15/24
Axisymmetry Tutorial 6-15
Phase2 v.8.0 Tutorial Manual
As you can see, the displacement levels off and becomes constant at a
certain distance away from the tunnel face. This curve is useful in that it
allows us to see the distance at which end effects can be ignored, and
plane strain conditions can be assumed. Also, this curve can be used to
estimate the “load split” factors, as described in the Support Tutorial,
Step 2.
Deleting a Query
Queries are deleted with the Delete Material Query option. However, a
convenient shortcut to delete a single query, is to simply right-click on the
query and select Delete Query.
1. First close the graph if you are still viewing it.
2. Right-click on the query and select Delete Query, and the query
will be removed from the model.
Graphing Multiple Queries
Now let’s create two more queries, this time perpendicular to the tunnel,
and plot them on the same graph.
Select: Query→ Add Material Query
1. The Snap option should still be in effect, so use the mouse to
select the vertex at (4 , -24), and then select the vertex at (28 ,
-24). Right-click the mouse and select Done.
2. In the Specify Query Locations dialog, enter 50 locations and
select OK.
Add another query.
Select: Query→ Add Material Query
1. Use the mouse to select the vertex at (4 , 0).
2. Now enter the coordinates (28 , 0) in the prompt line.
Alternatively, you can right-click and select the Ortho snap
option, which will allow you to snap the query exactly along a
horizontal line, and also to the external boundary (at the point
28,0) because the Snap option is also enabled. Right-click the
mouse and select Done.
3. In the Specify Query Locations dialog, enter 50 locations and
select OK.
You have created two new queries, one along the lower edge of the model,
and a parallel one at the face of the tunnel.
7/28/2019 Tutorial 06 Axisymmetric
http://slidepdf.com/reader/full/tutorial-06-axisymmetric 16/24
Axisymmetry Tutorial 6-16
Phase2 v.8.0 Tutorial Manual
This time, to graph the queries, we will use the Graph Material Queries
option, since we want both queries on the same graph. (The right-click
shortcut can only be used to graph a single query).
Select: Graph→ Graph Material Queries
1. Select the two queries by clicking on them with the left mousebutton. (Alternatively, you could right-click the mouse and choose
Select All from the popup menu.)
2. Right-click and select Graph Selected (or just press Enter), and
you will see the Graph Query Data dialog.
3. Select the Plot button in the dialog, and you should see the
following graph.
Figure 6-7: Total displacement perpendicular to tunnel, at face (lower curve), and at24 meters from face (upper curve).
The total displacement decreases as we move away from the tunnel. Note:
• the Total Displacement along the lower boundary is exactly
equivalent to the Horizontal (X) Displacement, since the Vertical
(Y) Displacement along this boundary is zero. (If you graphed this
query while viewing Horizontal Displacement instead of Total
Displacement, you could verify this for yourself.)
• The Total Displacement curve at the face of the tunnel includes
both X and Y displacement components.
Now close the graph view by selecting the X in the upper right corner of
the view.
7/28/2019 Tutorial 06 Axisymmetric
http://slidepdf.com/reader/full/tutorial-06-axisymmetric 17/24
Axisymmetry Tutorial 6-17
Phase2 v.8.0 Tutorial Manual
Plane Strain Comparison with Axisymmetric Results
To further illustrate the significance and meaning of an axisymmetric
model, we will create a plane strain model which is equivalent in allrespects to our axisymmetric model (except of course that the tunnel will
now be infinite, with no ‘end effects’), and compare the analysis results.
From Interpret, switch back to Model.
Select: Analysis→Model
(Or if you quit the program and are just re-starting the tutorial at this
point, then start the Phase2 Model program.)
Model
If you have been following this tutorial from the beginning, and the
axisymmetric model is still open:
Select: File→ New
to open a new document window, so that you can begin creating a new
model.
Plane strain is the default analysis type, so you do not have to set this in
Project Settings, it will already be in effect. Let’s first create a circular
tunnel of radius 4 meters (i.e., the same radius as the axisymmetric
tunnel).
Select: Boundaries→ Add Excavation
1. Right-click and select the Circle option from the popup menu.
2. In the Circle Options dialog, select the Center and radius option,
enter radius = 4, enter Number of segments = 60, and select OK.
7/28/2019 Tutorial 06 Axisymmetric
http://slidepdf.com/reader/full/tutorial-06-axisymmetric 18/24
Axisymmetry Tutorial 6-18
Phase2 v.8.0 Tutorial Manual
3. At the prompt, enter 0,0 as the circle center, and the circular
excavation will be created.
Now add the external boundary.
Select: Boundaries→ Add External
We will use the default parameters, so just select OK to automatically
create a BOX external boundary with an expansion factor of 3.
NOTE: this external boundary is the same distance away from the
excavation, as the right edge of the external boundary for the
axisymmetric problem (i.e. 28 meters from the center of the tunnel).
Mesh
Now discretize and mesh the model. First select Mesh Setup.
Select: Mesh→ Setup
In the Mesh Setup dialog, set the Number of Excavation Nodes to 60.
Select the Discretize button in the Mesh Setup dialog (this is equivalent
to using the Discretize option in the Mesh menu). The status bar will
indicate:
Di scr et i zati ons: Excavati on=60, Exter nal =68
Select the Mesh button in the Mesh Setup dialog (this is equivalent to
using the Mesh option in the Mesh menu). The status bar will indicate:
NODES = 697 ELEMENTS = 1324
Select OK in the Mesh Setup dialog.
Boundary Conditions
We will use the default boundary condition, which is a fixed (i.e. zero
displacement) condition on the external boundary. This corresponds to
the Fixed XY condition of the right edge of the external boundary in the
axisymmetric model.
7/28/2019 Tutorial 06 Axisymmetric
http://slidepdf.com/reader/full/tutorial-06-axisymmetric 19/24
Axisymmetry Tutorial 6-19
Phase2 v.8.0 Tutorial Manual
Field Stress
We will use the default Field Stress (i.e. hydrostatic conditions σ1=σ3=σZ
= 10MPa, which is the same field stress we used for the axisymmetric
problem), so you do not need to use the Field Stress option.
Properties
We will use the default rock material properties, so you do not have to
enter or assign any properties. The default properties are Material Type =
Elastic, Young’s Modulus = 20,000 MPa, Poisson’s ratio = 0.3.
However, we do have to excavate the tunnel. Let’s do this with a right-
click shortcut.
1. Right-click the mouse inside of the circular excavation.
2. In the popup menu, go to the Assign Material sub-menu, and
select the Excavate option.
3. The circular tunnel is now excavated.
The model should appear as below.
Figure 6-8: Infinite tunnel, 4 meter radius, plane strain problem.
Compute
Before you analyze your model, save it as a file called axi2.fez.
Select: File→ Save
7/28/2019 Tutorial 06 Axisymmetric
http://slidepdf.com/reader/full/tutorial-06-axisymmetric 20/24
Axisymmetry Tutorial 6-20
Phase2 v.8.0 Tutorial Manual
Use the Save As dialog to save the file. You are now ready to run the
analysis. (Alternatively, if you select Compute before saving a new file,
Phase2 will recognize this, and display the Save As dialog).
Select: Analysis→ Compute
The Phase2 Compute engine will proceed in running the analysis. Whencompleted, you will be ready to view the results in Interpret.
Interpret
We will now wrap up this tutorial with a few comparisons between the
axisymmetric and plane strain models we have created. Switch back to
Interpret.
Select: Analysis→ Interpret
View the total displacement contours.
Select:
Notice the maximum displacement displayed in the status bar.
Maxi mumTot al Di spl acement = 0. 002416 m
This is almost identical to the maximum displacement from the
axisymmetric problem (0.00246).
Now let’s use the Query and Graph options again to plot the displacement
vs. distance from the tunnel boundary.
Select: Query→ Add Material Query
1. Enter the point (4 , 0) at the first prompt and (28 , 0) at the
second prompt. Right-click and select Done, or just press Enter.
2. In the Specify Query Locations dialog, enter 50 locations, and
select OK.
3. Notice the query created from the right edge of the tunnel to the
right edge of the external boundary. Right-click on the query and
select Graph Data.
4. You will see the Graph Query Data dialog. Select the Plot button
and a graph of total displacement will immediately be generated.
You can see the displacement decreasing from the maximum at the
tunnel boundary, to zero at the fixed, external boundary.
7/28/2019 Tutorial 06 Axisymmetric
http://slidepdf.com/reader/full/tutorial-06-axisymmetric 21/24
Axisymmetry Tutorial 6-21
Phase2 v.8.0 Tutorial Manual
Finally, let’s compare this curve with the equivalent query from the
axisymmetric analysis.
1. Open the axisymmetric model (axi1.fez) in the Interpret
program.
2. Right-click on the lower query (at the bottom edge of the model)and select Graph Data. In the Graph Query Data dialog select the
Plot button to generate the graph.
3. Now tile the views by selecting Tile Windows from the toolbar.
Your screen should look similar to the following.
Figure 6-9: Comparison of displacements for axisymmetric and plane strain model.
The Total Displacement graphs from the two models are nearly identical.
• One graph represents the query along the lower edge of the
axisymmetric model.
• The other is the query added on the plane strain model.
This illustrates the relationship between the axisymmetric and plane
strain models – although the two models look very different, we can
extract the same results from either one.
7/28/2019 Tutorial 06 Axisymmetric
http://slidepdf.com/reader/full/tutorial-06-axisymmetric 22/24
Axisymmetry Tutorial 6-22
Phase2 v.8.0 Tutorial Manual
Addi tional Exercises
When you plotted the total displacement contours for the plane strain
tunnel model, you may not have noticed, but the contours begin to get
“square” as you get further from the tunnel (immediately around the
tunnel they are circular). The displacements are conforming to the shapeof the external boundary and the fixed boundary condition we imposed on
it.
Radial Mesh
For circular problems such as this one, there is a more appropriate
meshing option we could have used, called Radial meshing. Radial
meshing produces a symmetric, reproducible radial mesh for symmetric
problems such as this.
As an additional exercise, re-do the plane strain circular tunnel analysis,
with the following changes:
• After you add the excavation, DO NOT add the external
boundary, but select Mesh Setup instead.
• In the Mesh Setup dialog, toggle the Mesh Type to Radial, the
Element Type to 4 Noded Quadrilaterals, and enter the
#Excavation Nodes = 60. Note that for a Radial mesh, an
Expansion Factor for the external boundary is entered, rather
than a Gradation Factor.
• Discretize and Mesh. The external boundary will appear when the
radial mesh is generated.
• Carry out the analysis and data interpretation as before.
When you plot the displacement versus distance from the tunnel, you
should get nearly identical results as when you used the BOX external
boundary. However, the displacement contours are no longer distorted,
and are circular at any distance from the excavation.
Enter:
Mesh Type =Radial
Elem. Type =4 Node Quad
Expansion Factor =3
#Excavation Nodes =60
7/28/2019 Tutorial 06 Axisymmetric
http://slidepdf.com/reader/full/tutorial-06-axisymmetric 23/24
Axisymmetry Tutorial 6-23
Phase2 v.8.0 Tutorial Manual
Also note that quadrilateral elements in conjunction with Radial
meshing, are very efficient and give very good results.
Figure 6-10: Total displacement contours with radial mesh.
Distance of External Boundary from Excavation
As pointed out in other tutorials, the distance of the external boundary
from the excavation(s), and the boundary conditions we impose on it, are
very important.
1. Create another Radial mesh, as described above, except this time
use an Expansion Factor of 5 (in the Mesh Setup dialog).
2. Re-run the analysis and plot the displacement versus distance
from the tunnel (add a query from 4,0 to 44,0).
3. Export the query data to Excel for both radial mesh models
(expansion factor = 3 and expansion factor = 5). Note: you can
export query data by right-clicking on a query and selecting Copy
Data or right-click on a graph and select Plot in Excel.
4. In Excel plot the query data to obtain Figure 6-11.
You will see that our previous Expansion Factor of 3 was too low, becausedisplacements increase significantly when we move the fixed external
boundary farther from the tunnel.
7/28/2019 Tutorial 06 Axisymmetric
http://slidepdf.com/reader/full/tutorial-06-axisymmetric 24/24
Axisymmetry Tutorial 6-24
Figure 6-11: Moving the fixed external boundary farther from the excavation results inincreased displacements.
The displacements near the excavation are comparable, but diverge
towards the external boundary. For example, at about 18 meters from the
excavation, the displacement for the Expansion Factor = 5 curve is about
double the Expansion Factor = 3 curve (see Figure 6-11).
The restraining effect of a fixed external boundary should therefore
always be considered. When comparing with analytical solutions, as in
the Phase2 verification manual, it is very important to take this into
account.
One final suggested exercise:
Re-do the axisymmetric problem and move the right edge of the external
boundary over to 44 meters. This gives an equivalent distance from the
excavation as the plain strain model with an expansion factor of 5.
Compare results with the equivalent plain strain model.
Upper curveExpansion factor =5
Lower curveExpansion factor =3